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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 6A / a / Solution

For

F (n) = α n - β n^{3}

, the nonnegative equilibria are

n = 0, n = K := α / β .

Since

F^{'} (0) = α > 0, F^{'} (K) = α - 3 β K^{2} = - 2 α < 0,

the zero equilibrium is unstable and

K

is asymptotically stable.

Solved by gpt-5.6-sol high.

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